4. Find the length of the following contours. (a) z(t)=3e^2it+2 (-pi leqslant tleqslant pi ) z(t)=e^tcost+ie^tsint (-pi leqslant tleqslant pi )

Here's how to find the lengths of the given contours:**Understanding the Concepts*** **Contour:** A contour is a continuous curve in the complex plane.* **Arc Length Formula:** The length of a contour defined by a complex function *z(t)* from *t = a* to *t = b* is given by: ``` L = ∫[a, b] |z'(t)| dt ``` where *z'(t)* is the derivative of *z(t)* with respect to *t*.**Solving the Problems****(a) z(t) = 3e^(2it) + 2**1. **Find the derivative:** * z'(t) = 6ie^(2it)2. **Calculate the magnitude of the derivative:** * |z'(t)| = |6ie^(2it)| = 6|e^(2it)| = 6 (since |e^(2it)| = 1)3. **Integrate to find the arc length:** * L = ∫[-π, π] 6 dt = 6t |[-π, π] = 12π**Therefore, the length of the contour z(t) = 3e^(2it) + 2 is 12π.****(b) z(t) = e^t cos(t) + ie^t sin(t)**1. **Find the derivative:** * z'(t) = e^t(cos(t) - sin(t)) + ie^t(sin(t) + cos(t))2. **Calculate the magnitude of the derivative:** * |z'(t)| = √[(e^t(cos(t) - sin(t)))² + (e^t(sin(t) + cos(t)))²] * |z'(t)| = √[e^(2t)(cos²(t) - 2cos(t)sin(t) + sin²(t) + sin²(t) + 2sin(t)cos(t) + cos²(t))] * |z'(t)| = √[e^(2t)(2)] = √2 * e^t3. **Integrate to find the arc length:** * L = ∫[-π, π] √2 * e^t dt = √2 * e^t |[-π, π] = √2(e^π - e^(-π))**Therefore, the length of the contour z(t) = e^t cos(t) + ie^t sin(t) is √2(e^π - e^(-π)).**